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M04.1 · Limits and continuity

A free, illustrated module of 4 lessons and one interactive in M04 Calculus I: Single Variable: What a limit is; Limits by algebra; One-sided limits and infinity; Continuity and the intermediate value theorem.

M04.1 L1 · Core 4 lessons · about 100 min Live · in review

Part of M04 Calculus I: Single Variable in the Mathematics track.

Calculus rests on one idea: the value a function approaches near a point, even where the function cannot be computed at the point itself. These four lessons estimate limits from tables and graphs and then pin the idea down as a game of tolerances, find limits by algebra and by squeezing (ending with sin(x)/x, which the derivative of sine needs), follow limits from one side and out to infinity, where the sigmoid levels off at 0 and 1, and define continuity. The last lesson turns the intermediate value theorem into the bisection method, and shows why the step function cannot be trained by following slopes while ReLU can.

What you need first. Functions and graphs (M01.3), exponentials (M01.4), radians and the unit circle (M01.5), and statements about every case and some case (M01.6). The code uses Python functions, loops and lists and imports math, NumPy and SciPy, explaining each new call.

Lessons

1. What a limit is M04.1.1 25 min

   A limit is the value a function heads towards near a point, even where it has no value at the point itself. Estimate one from a table and a graph, then pin it down as a game of tolerances: for every output tolerance there is an input window.

2. Limits by algebra M04.1.2 25 min

   Limits pass through sums, products and quotients, so most limits come from substituting. When substitution gives 0/0, factor and cancel, rationalise, or squeeze: areas on the unit circle trap sin(x)/x and show that it tends to 1.

3. One-sided limits and infinity M04.1.3 25 min

   Approach a point from one side only: a limit exists exactly when the left and right limits agree, which fails at a step. Let the output or the input run off to infinity, meet vertical and horizontal asymptotes, and watch the sigmoid level off at 0 and 1.

4. Continuity and the intermediate value theorem M04.1.4 25 min

   A function is continuous at a point when its limit there exists and equals its value. Tell removable, jump and infinite breaks apart, see why ReLU can be trained by slopes and a step cannot, and turn the intermediate value theorem into the bisection method.

The interactive

M04.1-X1 · Limit window. Pick a function with a hole, a jump or a smooth curve, set an output tolerance and shrink the input window. The band and the window are drawn on the graph, and the read-out says whether the graph stays inside the band. At the hole and on the curve some window always works; at the jump no window does once the band is narrow. It lives in What a limit is.